Groups with $\mathsf A_\ell$-commutator relations

Abstract:

If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\operatorname {GL}(\ell , A)$ has root subgroups $U_\alpha$ and Weyl elements $n_\alpha$ for $\alpha$ from the root system of type $\mathsf A_{\ell - 1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \geq 4$ satisfying natural conditions, then there is a way to recover the ring $A$. A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring $\operatorname {M}(\ell , A),$ a nonunital associative ring with a well-behaved Peirce decomposition is provided.